Carbon nanotubes (CNTs) are used for everything from structural reinforcement to communication antennas. The article “Fast Mechanochemical Synthesis of Carbon Nanotube-Polyanaline Hybrid Materials” (J. Mater. Res. 2018: 1486–1495) reported the following data on y = electrical conductivity (S/cm), x₁ = multi-walled CNT weight (mg), x₂ = CN_x weight (mg), and x₃ = water volume (ml) from one experiment.

a. Obtain and interpret R² and s_e for the model with predictors x₁, x₂, and x₃.
b. Test for model utility using α = .05.
c. Does the data support the manufacturing goal of (relatively) consistent electrical conductivity across differing values of the experimental factors? Explain.

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y 1.337 4.439 4.512 4.153 1.727 2.415 3.008 3.869 2.140 2.025 3.407 2.545 4.426 2.863 2.096 2.006 X1 0.0 20.2 40.6 20.3 20.1 0.0 20.3 20.3 40.9 20.4 40.4 20.4 20.3 40.4 0.0 20.8 X2 20.1 0.0 20.5 40.4 20.1 40.4 20.1 20.3 40.4 20.2 0.0 20.1 0.0 20.3 20.4 40.6 33 22227 2 7 7 7 7 12 12 12 12